Strong convergence to operator-valued semicirculars

TL;DR

This paper develops a framework for the convergence of matrix models to operator-valued semicircular systems, introducing covariance laws and establishing conditions for strong convergence in various matrix models.

Contribution

It introduces covariance laws and provides new conditions for strong and weak convergence of matrix models to operator-valued semicircular systems.

Findings

Established conditions for convergence of Gaussian matrices to operator-valued semicircular systems.

Proved strong convergence for Gaussian band matrices with continuous cutoff.

Constructed matrix models for interpolated free group factors.

Abstract

We establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices $\eta = (\eta_{i,j})_{i,j \in I}$. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of $\eta_{i,j}$, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a $B$-valued semicircular family and generators of the base algebra $B$. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.